On the Bäcklund transform and the stability of the line soliton of the KP-II equation on ${\mathbb{R}}^2$

Abstract

We study the Miura map of the KP-II equation on ${\mathbb{R}}^2$ and the resulting Bäcklund transform, which adds a line soliton to a given solution. This work aims to develop a complementary approach to T. Mizumachi's method for the $L^2$-stability of the line soliton, which the potential for generalization to multisolitons. We construct the Bäcklund transform by classifying solutions of the Miura map equation close to a modulated kink; this translates into studying eternal solutions of the forced viscous Burgers' equation under distinct boundary conditions at $\pm \infty$. We then show that its range, when intersected with a small ball in $|D_x{|}^{1/2}{L}^2({\mathbb{R}}^2)\cap L^2({\mathbb{R}}^2)\cap ⟨y{⟩}^{0-}{L}^1({\mathbb{R}}^2)$, forms a codimension-1 manifold. We prove codimension-1 $L^2$-stability of the line soliton in the aforementioned weighted space as a corollary, providing the first stability result at sharp regularity. The codimension-1 condition in the range of the Bäcklund transform is an intrinsic property, and we conjecture that it corresponds to a known long time behavior of perturbed line solitons. The stability is expected to hold without this condition, as in Mizumachi's works. Finally, we show the construction of a multisoliton addition map for $(k,1)$-multisolitons, $k\ge 1$.